The most difficult part is handling parent-child relations correctly:
when a parent is applied, the children should be moved accordingly,
and when applying a child, it should not include transformation from
unapplied parents. All this requires walking bones as a tree, instead
of a flat list.

Limitation: Applying bones with non-uniform scaling without also applying
children will distort non-rest posing on said children for reasons related
to T54159 (basically, non-uniform scale plus rotation creates shear, and
Blender matrix decomposition utilities don't have tools to deal with it).